2—Infinite Series 462.12 What is the series expansion forcscx= 1/sinx? As in the previous problem, use your knowledge
of the sine series and the geometric series to get this result at least throughx^5. Note: the first term
inthisseries is 1 /x. Ans: 1 /x+x/6 + 7x^3 /360 + 31x^5 /15120 +···
2.13 The exact relativistic expression for the kinetic energy of an object with non-zero mass is
K=mc^2
(
γ− 1
)
where γ=
(
1 −v^2 /c^2
)− 1 / 2
andcis the speed of light in vacuum. If the speedvis small compared to the speed of light, find
an approximate expression forKto show that it reduces to the Newtonian expression for the kinetic
energy, but include the next term in the expansion to determine how large the speedvmust be in order
that this correction term is 10% of the Newtonian expression for the kinetic energy? Ans:v≈ 0. 36 c
2.14 Use series expansions to evaluate
lim
x→ 01 −cosx
1 −coshx
and lim
x→ 0sinkx
x
2.15 Evaluate using series; you will need both the sine series and the binomial series.
lim
x→ 0(
1
sin^2 x
−
1
x^2
)
Now do it again, setting up the algebra differently and finding an easier (or harder) way. Ans: 1 / 3
2.16 For some more practice with series, evaluate
lim
x→ 0(
2
x
+
1
1 −
√
1 +x
)
Ans: Check experimentally with a few values ofxon a pocket calculator.
2.17 Expand the integrand to find the power series expansion for
ln(1 +x) =
∫x0dt(1 +t)−^1
Ans: Eq. (2.4)2.18 (a) The error functionerf(x)is defined by an integral. Expand the integrand, integrate term
by term, and develop a power series representation forerf. For what values ofxdoes it converge?
Evaluateerf(1)from this series and compare it to the result of problem1.34. (b) Also, as further
validation of the integral in problem1.13, do the power series expansion of both sides of the equation
and verify the expansions of the two sides of the equation agree.
2.19 Verify that the combinatorial factormCnis really what results for the coefficients when you
specialize the binomial series Eq. (2.4) to the case that the exponent is an integer.